Question

Derive equation of terminal velocity in free fall condition?​

Answer 1

$\huge\mathfrak{Bonjour!!}$

$\huge\fcolorbox{black}{aqua}{Solution:-}$

✒ Wʜᴀᴛ ɪs Tᴇʀᴍɪɴᴀʟ Vᴇʟᴏᴄɪᴛʏ?!

⛄ ᴛᴇʀᴍɪɴᴀʟ ᴠᴇʟᴏᴄɪᴛʏ ɪs ᴛʜᴇ ᴍᴀxɪᴍᴜᴍ ᴠᴇʟᴏᴄɪᴛʏ ᴀᴛᴛᴀɪɴᴀʙʟᴇ ʙʏ ᴀɴ ᴏʙᴊᴇᴄᴛ ᴀs ɪᴛ ғᴀʟʟs ᴛʜʀᴏᴜɢʜ ᴀ ғʟᴜɪᴅ (ᴀɪʀ ɪs ᴛʜᴇ ᴍᴏsᴛ ᴄᴏᴍᴍᴏɴ ᴇxᴀᴍᴘʟᴇ). ɪᴛ ᴏᴄᴄᴜʀs ᴡʜᴇɴ ᴛʜᴇ sᴜᴍ ᴏғ ᴛʜᴇ ᴅʀᴀɢ ғᴏʀᴄᴇ (ғᴅ) ᴀɴᴅ ᴛʜᴇ ʙᴜᴏʏᴀɴᴄʏ ɪs ᴇϙᴜᴀʟ ᴛᴏ ᴛʜᴇ ᴅᴏᴡɴᴡᴀʀᴅ ғᴏʀᴄᴇ ᴏғ ɢʀᴀᴠɪᴛʏ (ғɢ) ᴀᴄᴛɪɴɢ ᴏɴ ᴛʜᴇ ᴏʙᴊᴇᴄᴛ. sɪɴᴄᴇ ᴛʜᴇ ɴᴇᴛ ғᴏʀᴄᴇ ᴏɴ ᴛʜᴇ ᴏʙᴊᴇᴄᴛ ɪs ᴢᴇʀᴏ, ᴛʜᴇ ᴏʙᴊᴇᴄᴛ ʜᴀs ᴢᴇʀᴏ ᴀᴄᴄᴇʟᴇʀᴀᴛɪᴏɴ.[1]

ɪɴ ғʟᴜɪᴅ ᴅʏɴᴀᴍɪᴄs, ᴀɴ ᴏʙᴊᴇᴄᴛ ɪs ᴍᴏᴠɪɴɢ ᴀᴛ ɪᴛs ᴛᴇʀᴍɪɴᴀʟ ᴠᴇʟᴏᴄɪᴛʏ ɪғ ɪᴛs sᴘᴇᴇᴅ ɪs ᴄᴏɴsᴛᴀɴᴛ ᴅᴜᴇ ᴛᴏ ᴛʜᴇ ʀᴇsᴛʀᴀɪɴɪɴɢ ғᴏʀᴄᴇ ᴇxᴇʀᴛᴇᴅ ʙʏ ᴛʜᴇ ғʟᴜɪᴅ ᴛʜʀᴏᴜɢʜ ᴡʜɪᴄʜ ɪᴛ ɪs ᴍᴏᴠɪɴɢ [2].

ᴀs ᴛʜᴇ sᴘᴇᴇᴅ ᴏғ ᴀɴ ᴏʙᴊᴇᴄᴛ ɪɴᴄʀᴇᴀsᴇs, sᴏ ᴅᴏᴇs ᴛʜᴇ ᴅʀᴀɢ ғᴏʀᴄᴇ ᴀᴄᴛɪɴɢ ᴏɴ ɪᴛ, ᴡʜɪᴄʜ ᴀʟsᴏ ᴅᴇᴘᴇɴᴅs ᴏɴ ᴛʜᴇ sᴜʙsᴛᴀɴᴄᴇ ɪᴛ ɪs ᴘᴀssɪɴɢ ᴛʜʀᴏᴜɢʜ (ғᴏʀ ᴇxᴀᴍᴘʟᴇ ᴀɪʀ ᴏʀ ᴡᴀᴛᴇʀ). ᴀᴛ sᴏᴍᴇ sᴘᴇᴇᴅ, ᴛʜᴇ ᴅʀᴀɢ ᴏʀ ғᴏʀᴄᴇ ᴏғ ʀᴇsɪsᴛᴀɴᴄᴇ ᴡɪʟʟ ᴇϙᴜᴀʟ ᴛʜᴇ ɢʀᴀᴠɪᴛᴀᴛɪᴏɴᴀʟ ᴘᴜʟʟ ᴏɴ ᴛʜᴇ ᴏʙᴊᴇᴄᴛ (ʙᴜᴏʏᴀɴᴄʏ ɪs ᴄᴏɴsɪᴅᴇʀᴇᴅ ʙᴇʟᴏᴡ). ᴀᴛ ᴛʜɪs ᴘᴏɪɴᴛ ᴛʜᴇ ᴏʙᴊᴇᴄᴛ ᴄᴇᴀsᴇs ᴛᴏ ᴀᴄᴄᴇʟᴇʀᴀᴛᴇ ᴀɴᴅ ᴄᴏɴᴛɪɴᴜᴇs ғᴀʟʟɪɴɢ ᴀᴛ ᴀ ᴄᴏɴsᴛᴀɴᴛ sᴘᴇᴇᴅ ᴄᴀʟʟᴇᴅ ᴛʜᴇ ᴛᴇʀᴍɪɴᴀʟ ᴠᴇʟᴏᴄɪᴛʏ (ᴀʟsᴏ ᴄᴀʟʟᴇᴅ sᴇᴛᴛʟɪɴɢ ᴠᴇʟᴏᴄɪᴛʏ). ᴀɴ ᴏʙᴊᴇᴄᴛ ᴍᴏᴠɪɴɢ ᴅᴏᴡɴᴡᴀʀᴅ ғᴀsᴛᴇʀ ᴛʜᴀɴ ᴛʜᴇ ᴛᴇʀᴍɪɴᴀʟ ᴠᴇʟᴏᴄɪᴛʏ (ғᴏʀ ᴇxᴀᴍᴘʟᴇ ʙᴇᴄᴀᴜsᴇ ɪᴛ ᴡᴀs ᴛʜʀᴏᴡɴ ᴅᴏᴡɴᴡᴀʀᴅs, ɪᴛ ғᴇʟʟ ғʀᴏᴍ ᴀ ᴛʜɪɴɴᴇʀ ᴘᴀʀᴛ ᴏғ ᴛʜᴇ ᴀᴛᴍᴏsᴘʜᴇʀᴇ, ᴏʀ ɪᴛ ᴄʜᴀɴɢᴇᴅ sʜᴀᴘᴇ) ᴡɪʟʟ sʟᴏᴡ ᴅᴏᴡɴ ᴜɴᴛɪʟ ɪᴛ ʀᴇᴀᴄʜᴇs ᴛʜᴇ ᴛᴇʀᴍɪɴᴀʟ ᴠᴇʟᴏᴄɪᴛʏ. ᴅʀᴀɢ ᴅᴇᴘᴇɴᴅs ᴏɴ ᴛʜᴇ ᴘʀᴏᴊᴇᴄᴛᴇᴅ ᴀʀᴇᴀ, ʜᴇʀᴇ, ᴛʜᴇ ᴏʙᴊᴇᴄᴛ's ᴄʀᴏss-sᴇᴄᴛɪᴏɴ ᴏʀ sɪʟʜᴏᴜᴇᴛᴛᴇ ɪɴ ᴀ ʜᴏʀɪᴢᴏɴᴛᴀʟ ᴘʟᴀɴᴇ. ᴀɴ ᴏʙᴊᴇᴄᴛ ᴡɪᴛʜ ᴀ ʟᴀʀɢᴇ ᴘʀᴏᴊᴇᴄᴛᴇᴅ ᴀʀᴇᴀ ʀᴇʟᴀᴛɪᴠᴇ ᴛᴏ ɪᴛs ᴍᴀss, sᴜᴄʜ ᴀs ᴀ ᴘᴀʀᴀᴄʜᴜᴛᴇ, ʜᴀs ᴀ ʟᴏᴡᴇʀ ᴛᴇʀᴍɪɴᴀʟ ᴠᴇʟᴏᴄɪᴛʏ ᴛʜᴀɴ ᴏɴᴇ ᴡɪᴛʜ ᴀ sᴍᴀʟʟ ᴘʀᴏᴊᴇᴄᴛᴇᴅ ᴀʀᴇᴀ ʀᴇʟᴀᴛɪᴠᴇ ᴛᴏ ɪᴛs ᴍᴀss, sᴜᴄʜ ᴀs ᴀ ʙᴜʟʟᴇᴛ. ɪɴ ɢᴇɴᴇʀᴀʟ, ғᴏʀ ᴛʜᴇ sᴀᴍᴇ sʜᴀᴘᴇ ᴀɴᴅ ᴍᴀᴛᴇʀɪᴀʟ, ᴛʜᴇ ᴛᴇʀᴍɪɴᴀʟ ᴠᴇʟᴏᴄɪᴛʏ ᴏғ ᴀɴ ᴏʙᴊᴇᴄᴛ ɪɴᴄʀᴇᴀsᴇs ᴡɪᴛʜ sɪᴢᴇ. ᴛʜɪs ɪs ʙᴇᴄᴀᴜsᴇ ᴛʜᴇ ᴅᴏᴡɴᴡᴀʀᴅ ғᴏʀᴄᴇ (ᴡᴇɪɢʜᴛ) ɪs ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟ ᴛᴏ ᴛʜᴇ ᴄᴜʙᴇ ᴏғ ᴛʜᴇ ʟɪɴᴇᴀʀ ᴅɪᴍᴇɴsɪᴏɴ, ʙᴜᴛ ᴛʜᴇ ᴀɪʀ ʀᴇsɪsᴛᴀɴᴄᴇ ɪs ᴀᴘᴘʀᴏxɪᴍᴀᴛᴇʟʏ ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟ ᴛᴏ ᴛʜᴇ ᴄʀᴏss-sᴇᴄᴛɪᴏɴ ᴀʀᴇᴀ ᴡʜɪᴄʜ ɪɴᴄʀᴇᴀsᴇs ᴏɴʟʏ ᴀs ᴛʜᴇ sϙᴜᴀʀᴇ ᴏғ ᴛʜᴇ ʟɪɴᴇᴀʀ ᴅɪᴍᴇɴsɪᴏɴ. ғᴏʀ ᴠᴇʀʏ sᴍᴀʟʟ ᴏʙᴊᴇᴄᴛs sᴜᴄʜ ᴀs ᴅᴜsᴛ ᴀɴᴅ ᴍɪsᴛ, ᴛʜᴇ ᴛᴇʀᴍɪɴᴀʟ ᴠᴇʟᴏᴄɪᴛʏ ɪs ᴇᴀsɪʟʏ ᴏᴠᴇʀᴄᴏᴍᴇ ʙʏ ᴄᴏɴᴠᴇᴄᴛɪᴏɴ ᴄᴜʀʀᴇɴᴛs ᴡʜɪᴄʜ ᴘʀᴇᴠᴇɴᴛ ᴛʜᴇᴍ ғʀᴏᴍ ʀᴇᴀᴄʜɪɴɢ ᴛʜᴇ ɢʀᴏᴜɴᴅ ᴀɴᴅ ʜᴇɴᴄᴇ ᴛʜᴇʏ sᴛᴀʏ sᴜsᴘᴇɴᴅᴇᴅ ɪɴ ᴛʜᴇ ᴀɪʀ ғᴏʀ ɪɴᴅᴇғɪɴɪᴛᴇ ᴘᴇʀɪᴏᴅs. ᴀɪʀ ᴘᴏʟʟᴜᴛɪᴏɴ ᴀɴᴅ ғᴏɢ ᴀʀᴇ ᴇxᴀᴍᴘʟᴇs ᴏғ ᴛʜɪs.

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⛄ ∂єriviทg τєrмiทαℓ vєℓσciτy ƒσr τнє ƒrєє ƒαℓℓ σƒ αท σвjєcτ υsiทg мατнємατicαℓ τєrмs αท∂ єqυατiσทs αrє αs ƒσℓℓσωs:-

★ v = vєℓσciτy

★ ρ = gαs ∂єทsiτy

★ α = ƒrσทταℓ αrєα

★ c∂ = ∂rαg cσєƒƒiciєทτ

ทєτ. ƒσrcє єqυαℓs ∂rαg мiทυs ωєigнτ.

➻ ƒ - ∂ = ω

∂rαg єqυατiσท:-

➻ ∂ = c∂ ρv²α/2

∂rαg iทcrєαsєs ωiτн τнє sqυαrє σƒ τнє vєℓσciτy.

ωнєท ∂rαg is єqυαℓ τσ ωєigнτ τнєrє is ทσ ทєτ ƒσrcє σท τнє σвjєcτ.

➻ ƒ = ∂ - ω = 0

τнєท,

➻ c∂ ρv²α/2 = ω

∴ τєrмiทαℓ vєℓσciτy:-

➻ v = √(2ω/c∂ ρα)

cσмραriทg τωσ σвjєcτs, τнє нigнєr vєℓσciτy σccυrs ƒσr grєατєr ωєigнτ, ℓσωєr ∂rαg cσєƒƒiciєทτ (мσrє sτrєαмℓiทє∂), ℓσωєr gαs ∂єทsiτy (нigнєr αℓτiτυ∂є), σr sмαℓℓєr αrєα.

• σвjєcτs ∂σ ทστ ƒαℓℓ ατ τнє sαмє rατє τнrσυgн τнє ατмσsρнєrє.

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